18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.12 Generating Functions 18.14 Inequalities

§18.13 Continued Fractions

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Keywords:: classical orthogonal polynomials, continued fractions
Notes:: See Lorentzen and Waadeland (1992, pp. 446–448).
Permalink:: http://dlmf.nist.gov/18.13
See also:: Annotations for Ch.18

We use the terminology of §1.12(ii). The following formulae are explicit cases of (18.2.34)–(18.2.36); this area is fully explored in §§18.30(vi) and 18.30(vii).

Chebyshev

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Keywords:: Chebyshev polynomials, continued fractions
See also:: Annotations for §18.13 and Ch.18

$T_{n}\left(x\right)$ is the denominator of the $n$ th approximant to:

18.13.1		$\cfrac{-1}{x+\cfrac{-1}{2x+\cfrac{-1}{2x+}}}\cdots,$
ⓘ Symbols: $x$ : real variable Permalink: http://dlmf.nist.gov/18.13.E1 Encodings: TeX, pMML, png See also: Annotations for §18.13, §18.13 and Ch.18

and $U_{n}\left(x\right)$ is the denominator of the $n$ th approximant to:

18.13.2		$\cfrac{-1}{2x+\cfrac{-1}{2x+\cfrac{-1}{2x+}}}\cdots.$
ⓘ Symbols: $x$ : real variable Permalink: http://dlmf.nist.gov/18.13.E2 Encodings: TeX, pMML, png See also: Annotations for §18.13, §18.13 and Ch.18

Legendre

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Keywords:: Legendre polynomials, continued fraction
See also:: Annotations for §18.13 and Ch.18

$P_{n}\left(x\right)$ is the denominator of the $n$ th approximant to:

18.13.3		$\cfrac{a_{1}}{x+\cfrac{-\frac{1}{2}}{\frac{3}{2}x+\cfrac{-\frac{2}{3}}{\frac{5% }{3}x+\cfrac{-\frac{3}{4}}{\frac{7}{4}x+}}}}\cdots,$
ⓘ Symbols: $x$ : real variable Permalink: http://dlmf.nist.gov/18.13.E3 Encodings: TeX, pMML, png See also: Annotations for §18.13, §18.13 and Ch.18

where $a_{1}$ is an arbitrary nonzero constant.

Laguerre

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Keywords:: Laguerre polynomials, continued fraction
See also:: Annotations for §18.13 and Ch.18

$L_{n}\left(x\right)$ is the denominator of the $n$ th approximant to:

18.13.4		$\cfrac{a_{1}}{1-x+\cfrac{-\frac{1}{2}}{\frac{1}{2}(3-x)+\cfrac{-\frac{2}{3}}{% \frac{1}{3}(5-x)+\cfrac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}}}}\cdots,$
ⓘ Symbols: $x$ : real variable Permalink: http://dlmf.nist.gov/18.13.E4 Encodings: TeX, pMML, png See also: Annotations for §18.13, §18.13 and Ch.18

where $a_{1}$ is again an arbitrary nonzero constant.

Hermite

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Keywords:: Hermite polynomials, classical orthogonal polynomials, continued fractions
See also:: Annotations for §18.13 and Ch.18

$H_{n}\left(x\right)$ is the denominator of the $n$ th approximant to:

18.13.5		$\cfrac{1}{2x+\cfrac{-2}{2x+\cfrac{-4}{2x+\cfrac{-6}{2x+}}}}\cdots.$
ⓘ Symbols: $x$ : real variable Permalink: http://dlmf.nist.gov/18.13.E5 Encodings: TeX, pMML, png See also: Annotations for §18.13, §18.13 and Ch.18

See also Cuyt et al. (2008, pp. 91–99).

18.12 Generating Functions 18.14 Inequalities

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